Use permissions.request to unify request handling.

[lgarron]

@jyasskin, is https://github.com/chromium/permissions.request "ready for production"? Can/should we use it?

[jyasskin]

You'd be the first production user, and the code has no tests, so beware. Aside from the lack of tests, though, I think it's close, so if you're willing to help remove bugs, I'd be happy to get a user.

[jyasskin]

Also note that the interface doesn't quite match https://w3c.github.io/permissions/: writing the polyfill revealed some things we'll need to fix.